Romanian Reports in Physics, Vol. 58, No. 4, P. 519–528, 2006

NONLINEARITY IN DESORPTIONFROM METALLIC SURFACES. I

D. BEJAN

University of Bucharest, Faculty of Physics, Department of Optics, P.O. Box MG 11, Bucharest,Romania, e-mail:dbejan@easynet.ro

(Received November 11, 2004)

Abstract. Photodesorption of molecular adsorbates from the metallic surfaces was studied bywave packet propagation of the Schrödinger equation using a two state, conservative system. Thesystem Hamiltonian includes explicitely the laser electric field and the interaction terms arecalculated function of the desorption coordinate. Applied to photodesorption of CO on Cu(111) weobtained a quadratic yield/fluence dependence for time-dependent fields and a Y ≈ F4 dependence forstatic (but greater) fields.

Key words: desorption, nonlinearity, surface state, conservative system, direct excitation.

1. INTRODUCTION

The laser induced desorption from metallic surfaces exhibits some generalproperties like: i) a picosecond response time for desorption; ii) a nonthermalrotational population that can be converted to a very high mean rotationaltemperature and also a very high vibrational temperature; iii) low translationalexcitation; iv) a nonlinear dependence of the desorption yield on the laser fluence.These observations lead to a desorption process driven by the high degree of thesubstrate electronic excitation. A nonlinear yield/fluence dependence was alsoobtained in desorption induced by the STM (Scanning Tunneling Microscopy) tipand appears to be the most outstanding characteristic of the desorption process.

The mechanism of photodesorption almost generally adopted is close to theideas of Gadzuk [1]. The laser excites first the electrons of the metal creating a bathof hot, non-equilibrium, electrons that will scatter into an unoccupied valenceelectron resonance of the adsorbate forming a temporary negative ion. After theneutralization of this ion, the system returns to the ground state of the adsorbateand an excited state of the substrate located in the conduction band of the metal.During the lifetime of the adsorbate resonance some energy is transferred to thedesorption coordinate leading to a single excitation event (desorption induced byelectronic transition, DIET) or multiple-excitations event (desorption induced by

520 D. Bejan 2

multiple electronic transition, DIMET) inducing desorption. The reader is referredto the review of Guo, Saalfrank and Seideman [2] and also to our recent review [3]for an extensive overview of concepts and theoretical models employed in the fieldof photodesorption.

An experiment on STM-induced CO desorption from Cu(111) [4] showedthat the desorption is due to the attachment of a single electron into the excited 2π*orbital of CO. Also a recent two-photon photoemission spectroscopy experimentperformed on CO/Cu(111) [5] showed that the 2π* orbital of CO may be populatedby direct and indirect, substrate mediated, excitations. These experiments raise thequestion of the importance of the interplay between these direct and indirectexcitations of the adsorbate resonance in the desorption mechanism. They alsoindicate that the specific surface cut of the metallic substrate and the associatedsurface projected band structure can influence the desorption process. Up to now,the direct excitation by the electric field of the laser radiation was taken intoaccount only in few papers [6–9] and the metallic surface is always takenstructureless. Consequently the most common models for photodesorption retainonly the gas phase characteristics of the adsorbate-substrate system.

The present paper is based on a phenomenological approach. We search forthe ingredients of a theoretical model giving rise to the nonlinear behaviour of thephotodesorption yield with the laser fluence. We used a diabatic model, composedof the standard two states potential energy curves (PECs) and the associatedinterstate interactions, in a time dependent, based on the Schrödinger equationformalism. We explicitly calculated the dependence of the interaction terms on thedesorption coordinate. In the preceeding phenomenological models, many differentexpressions of the interaction terms, deduced from qualitative arguments, wereproposed but, so far, there was no explicit attempt to calculate these dependences.For example, a Lorentzian form [6], a Gaussian form [10, 11], an exponential formbased on analogy with optical transitions [12], a form derived from a Boltzmannfactor [13, 14] and finally a decaying exponential form [15] were already used.

We apply the present phenomenological models to a CO molecule adsorbedon a Cu(111) substrate, a system we studied before [11, 16] that is rather wellcharacterised both experimentally [5, 17, 18] and theoretically [8, 9, 11, 16, 19–22].In the calculations presented here we introduce the structure of Cu(111) surfacethat at Γ presents a discrete ground surface state (SS) below the Fermi level.

2. TWO-STATE MODEL

The dynamics of the motion of the nuclei occurs on the one-dimensionaldiabatic potential energy curves (PECs) presented in Fig. 1. The coordinate Z isthe desorption coordinate and it describes the motion of the CO adsorbate center of

3 Nonlinearity in desorption. I 521

Fig. 1 – Potential energy curves for desorption of CO from Cu(111).

mass (CoM) normal to the surface. The PECs are obtained as in our previouspapers [11, 16] from the potential of Tully [23] and the charge transferparameterisation of the negative ion state. The curve V1 corresponds to the groundstate 1 of the system built from the neutral molecule bonded to the neutral metal.

The curve V2 corresponds to the state 2 , a charge transfer from the metal to the

empty 2π* orbital of CO molecule, giving rise to the negative ion resonance.The Cu(111) projected band structure at Γ point (k|| = 0) presents a band

gap extending between –0.85 and 4.3 eV and a surface state (SS) located at ESS = –0.39 eV [24, 25]. So, for k|| = 0, at the Fermi level, there are no available bulkcontinuum electronic states, the last occupied state being the SS. We chose the SSas the ground electronic state of 1 and we associate the nuclei V1 PEC with it.The zero energy of the adsorbate-substrate model is associated with the desorptionasymptote of V1.

In the matrix form, the Schrödinger equation for the nuclear motion in thedesorption coordinate Z on the two diabatic potential curves is:

1 1 12 12 1

2 12 12 2 2

( , ) ( , ) ( ) ( ,

( , ) ( , ) ( ) ( , )

Z t H V Z t Q Z Z tdidt Z t V Z t Q Z H Z t

ψ + ψ⎛ ⎞ ⎛ ⎞⎛ ⎞=⎜ ⎟ ⎜ ⎟⎜ ⎟ψ + ψ⎝ ⎠ ⎝ ⎠⎝ ⎠

(1)

In this equation the electronic coordinates are already integrated out. Each Hi

describes the motion of the nuclei on an uncoupled diabatic PEC Vi , [ ]1, 2 ,i∈

2 2

22i idH V

dZ= − +

μ (2)

The two terms above correspond to the kinetic and potential energy operatorsof the Hamiltonian for the motion of the nuclei and μ is the reduced mass of the

522 D. Bejan 4

system. Q12 is the electrostatic interaction between the diabatic states 1 and 2that determines the non-radiative electron charge transfer, taken to be symmetric.The direct excitation/de-excitation of the ion state 2 by the laser field is describedby the symmetric interaction term V12(Z, t):

// // 2 2 212 0 012

1( , ) ( ) cos( ) ( ) ( ) cos ( ) ( )2

k kxxV Z t e D Z E t g t Z E wt g t⎡ ⎤= − ω + α⎢ ⎥⎣ ⎦

(3)

where E0 is the modulus of the electric field vector, ω is the light angular frequency,//

12 ( )kD Z is the electric dipole moment for the transition between states 1 and 2

and //kxxα accounts for the polarisability term. The second term becomes important

at the laser fluences used here. The laser pulse temporal form g(t) is taken to be ofa Gaussian shape g(t) = exp(–(t – tp)2 2ln2 / ),pτ with 2τp the full width at half-

maximum (FWHM).Similar laser-matter coupling, including terms beyond the dipole moments,

were used for desorption of NH3 from a copper surface [26]. However, in thementioned work, the dipole moment is taken from the gas phase work and theinfluence of the substrate is neglected. In our two-state model the influence of the

substrate is partly taken into account through //12 ( )kD Z .

In the phenomenological photodesorption models, one usually takes [6, 7] the

dipole transition moment //12 ( )kD Z as independent of the desorption coordinate Z

and this approximation can artificially modify the results. In fact the transitiondipole moment is proportional to the overlap between the metal and adsorbateelectronic wave functions, overlap that tends to zero when the adsorbate-substratedistance increases at desorption.

In a first approximation, the transition dipole moment can be taken as theinteraction between the SS and the 2π* electronic wave functions, corresponding toV1 and V2 PECs, as we already did in our previous paper [16]:

//

//

2 *12 ( ) ( , , ) ( , , , )k SS

kD z x y z x x y z Zπ= φ φ − (4)

the integration is performed over the electronic coordinates (x, y, z) and the integralis parametrical dependent on Z, the nuclear desorption coordinate. In thecalculation of the integrals we have to deal with two parallel electronic coordinatesystems: the first centered at the Cu atom with z perpendicular on the metal surfaceand the second centered on the CO molecule with the z origin shifted by Z, thedistance between the CO-CoM (CoM is the center of mass) and the surface.

For the //

( , , )SSk x y zφ wave function we used, following Goodwin [27], the

nearly free electron approximation. The corresponding wave function [16] is a

5 Nonlinearity in desorption. I 523

traveling plane wave in the x and y directions but it is quickly damped in the zdirection in the bulk (z < 0) as well as in the vacuum (z > 0):

//

// //

0 // //

exp( )cos( ) 0( , , )

exp( ) 0

sSS sk

s

zA qz ik r for zax y z

B z ik r for z

π⎧ + + δ <⎪φ = ⎨⎪ −γ + ≥⎩

(5)

where As, Bs are the normalization factors, k|| is the wave vector for the free motionin the x and y directions (abbreviated as r||) and the three factors q, α, δ arecorrelated and dependent on the SS periodic pseudo-potential that models the effectof the screened cores of the solid [16].

The CO 2π* resonance, of total 2Π symmetry, can be simply represented by alinear combination of only two partial waves pπ and dπ centered on the CO-CoM[28]:

2 *11 21ˆ ˆ( , , ) ( ) ( ) ( ) ( )p p d dx y z A R r Y r A R r Y rπφ = + (6)

where Ap and Ad are normalization constants, Rp and Rd are Gaussian radialfunctions for the pπ and dπ waves, and Y11, Y21 are spherical harmonics centered atCO-CoM.

The dipole matrix element of eq. (4) is calculated by numerical integrationand the result is presented in Fig. 2. The integral is influenced by the exponents ofthe Gaussian functions for the pπ/dπ partial wave ratio and also by k||. In Fig. 2, themaximum value of the dipole momentum was set to one, in order to make a clearcomparison, for different k||, of the Z dependence of the transition dipole moment.This figure shows that for Z > 1.0 A°, region covered by our spatial grid used in thepropagation, the Z-dependence is essentially of the form

//012 ( ) exp( )kD Z d Z= −γ (7)

Fig. 2 – The transition dipole moment//

12 ( )kD Z as function of Z, the CO-surfacedistance, for different k|| values. Themaximum value of the transition moment is arbitrarily set to 1.

524 D. Bejan 6

for both k|| = 2π/as and k|| = π/as (here as is the interlayer spacing in the (111)direction for Cu). For k|| = 0 the Z-dependence is slightly more complicated.

To calculate the polarizability term // ( )kxx Zα of eq. (3) we performed an

integral similar to the eq. (4) where we replaced x by x2. The Z dependence of // ( )kxx Zα

is again function of the k|| value. For k|| = 2π/as and k|| = π/as one finds, as for the

dipole moment, an exponentially decaying dependence //0( ) exp( )k

xx Z Zα = α −γ thatwill be used in our numerical simulations.

To calculate the electrostatic interaction Q12(Z) we use an approximation dueto Gadzuk [29]. Namely, the interaction between the initial metallic surface and thefinal ion adsorbate states, corresponding to the resonant charge transfer, can beobtained using the Coulomb potential interaction operator centered on the ad-atomor the CoM of the CO molecule.

//

22 *

120

( ) ( , , ) ( , , , )4

SSk

eQ Z x y z x y z Zr

π−= φ φ −πε

(8)

Performing such numerical calculations, with the same parameters as for thedipole moment, one obtains Q12(Z) = 0 for k|| = 0. More elaborate calculations,including the interaction between the electronic configurations, would probablygive a non zero result for k|| = 0. For k|| = 2π/as and k|| = π/as, an exponentially

decaying Z-dependence, Q12(Z) = q0exp(–γZ) is obtained, where q0 is a function of

k|| and of the exponents of the Gaussian functions of the dπ/pπ partial waves. Thisexponentially decaying form, similar to the one found for the dipole momenteq. (7), will be used in the Hamiltonian matrix, eq. (1), for the propagation of theSchrödinger equation.

All the parameters for 2 *( , , )x y zπφ and //

( , , )SSk x y zφ wave functions are given

in Table 1. The values for the parameters d0, q0, α0 and γ are calculated for

k|| = π/as, i.e., at the boundary of the first Brillouin zone.The time dependent Schrödinger of eq. (1) is solved by space and time

propagation using a third-order split-operator (SO) method [30, 31]. At t0 we

assume that the system is entirely in the state 1 described by 1 0( , )Z tψ the

ground vibrational eigen function of the Hamiltonian H1 given by eq. (2). Thisstate represents the neutral molecule bonded to the metal with an initial populationp1 = 1. The end of the propagation is decided when the desorption probabilitystabilizes, reaching a plateau value. The desorption probability is obtained in thestandard way from the norm of the asymptotic wave function [11]:

7 Nonlinearity in desorption. I 525

max

0

2( ) ( , ) ,Z

des A

Z

P t Z t dZ= ψ∫ (9)

where Z0 and Zmax are the beginning and the final points of our spatial grid (seeTable 1).

Table 1

parameter value parameter value

ω 3.03 rad/fs τp 50 fs

σ 3.00 Å–1 as 2.08 A°

As 1.967 Å–1/2 Bs 1.529 Å–1/2

γ 1.005 Å–1 q 0.216 Å–1

Ap 0.375 Ad 0.625

d0 0.026 Å q0 0.052 eV

α0 200.00 Å2/V δ 0.68 rad

Z0 1.0 Å Zmax 21.47 Å

3. RESULTS

The phenomenological model developed above was applied to the study ofthe photodesorption of CO from Cu(111) for a laser of 2 eV, 100 fs and 45 J/m2

fluence, as in the experiment of Prybyla et al. [18]. In that experiment, theobserved quantum efficiency is of 10–3 molecules for a photon and will becompared to our desorption yield.

The total simulation time was 819.2 fs with a time step of Δt = 0.1 fs, chosento be small enough to resolve the variations in the Hamiltonian evolution and theoscillations of the external photon field. The spatial Z-grid had 2048 points spacedby an interval ΔZ = 0.01A°, so during the propagation time the wave packets neverreach the end of the grid and therefore no artificial oscillations will occur.

We display in Fig. 3 the dependence of the desorption yield as function of thelaser fluence for fluences between 30 J/m2 and 60 J/m2 as explored in theexperiment of Prybyla et al. [18]. The obtained yield ranging between 1⋅10–4 and5⋅10–4 is slightly lower than the experimental value of 10–3. We obtained ayield-fluence dependence linear or quadratic, function of the particular form usedfor the interaction term. Precisely, it is linear when only the dipole term of eq. (3)is introduced, quadratic when the polarisation term is added. We performed manyother calculations varying the parameters in the dipole transition moment eq. (4)and the electrostatic coupling eq. (8), without being able to enhance the yield-fluence

526 D. Bejan 8

dependence. The above result is obvious because, the system is conservative andonly the interaction term V12 is function of the laser fluence.

Let us now discuss in greater detail the present two-state model. At a givenfluence, the functional form of the transition moments and the electrostaticcouplings relative to the desorption coordinate Z, as well as the parameters used tocalculate these interactions, significantly influence the desorption yield. Howeverthey have only a minor impact on the yield-fluence dependence. Namely in Fig. 3all the parameters depend exponentially on Z, the desorption coordinate. Changingthis functional form does not improve the results. For example, taking a Gaussianform for Q12 and conserving the amplitude parameter q0, the recalculated desorptionyield is 10–5 times smaller but it still scales as F2. Moreover if, as in the publishedmodels [6, 7], we replaced the present exponentially decaying transition momentwith a constant value, we obtained yields 10 times greater but again a quadraticdependence of the desorption yield with the laser fluence.

Fig. 3 – Desorption yield for thetwo-state model as a function of thelaser fluence for the linear andquadratic laser-matter interaction V12.

The influence of changing k|| was also considered. We performed a calculationwith the Z-dependence of the transition moment corresponding to k|| = 0. For this k||

the transition moment first oscillates before decaying with an exponential form (seeFig. 2). We obtained only a lower desorption yield, due to the compensation of thepositive and negative lobes of the oscillatory part of the transition moment.

We varied also the maximum amplitudes of the dipole transition moment, ofthe polarisability and of the electrostatic coupling terms. Depending on theirparticular values, one can obtain a yield dependence on the fluence lower than theone presented in this paper, falling to a sub-linear dependence.

One observes that for k|| = π/as, the electrostatic couplings, the dipoletransition moment and polarizability transition moment decay exponentially withthe desorption coordinate Z. The interactions sum up and can be written as a uniqueanalytic expression:

9 Nonlinearity in desorption. I 527

212 12 0 0 0

1( ) ( , ) exp( ) ( ) ( )2

Q Z V Z t Z q ed E t e E t⎡ ⎤+ = −γ − − α⎢ ⎥⎣ ⎦ (10)

Due to this peculiarity, if one uses a static field, instead of the oscillating electricfield of a laser, the enhancement of the yield can be important. Depending on theparticular value of the parameters, one can obtain in this case a nonlinear fluencedependence of the yield, up to F4. This particular result was obtained with theparameters q0 = 0.2 eV, d0 = 4.5 A° and α0 = 200 A2/V and a static fields of modulusE0 = 1 V/A°. This electric field value is 100 times larger than the electric field ofthe laser used here. However, such fields are currently applied in STM and thereforean explanation of the observed nonlinear desorption induced by STM tip [4] can begiven by our model.

4. CONCLUSION

The conservative two-state model, composed of a ground and an ionic state,is the most widely used in the study of laser induced desorption [6–9, 12–15, 21,22, 32–34]. Our two-state model includes the direct excitation/de-excitation by thelaser and neglects the interaction with the metal bath. For an oscillating electricfield of the photon, the highest dependence of the observable on the laser fluence isF2. If instead one uses a static electric field about 100 times stronger, thisdependence becomes super-linear, of the order of F4. For NO/Pt(111), only a linearyield/fluence dependence was obtained by Saalfrank [7]. These results clearlyshow that the two-state model is unable to account for the complex indirectphenomena taking place upon desorption at metallic surfaces. The experimentaland theoretical evidence show that the direct excitation of the negative ion resonanceis not the main process and that the interaction between the adsorbate and the bathcontributes substantially to the desorption dynamics. A similar conclusion was alsoreached by Saalfrank [6] in the past but from a different point of view.

Supplementary ingredients are needed to account for the complex dynamicsof the electrons in the laser-adsorbate-substrate system. The published two-statequantum models introduce the hot electrons influence only through interactionterms that depend on an electronic temperature. This temperature appears to be theonly parameter able to induce the non-linearity in the desorption models. However,we already showed [35] that an electronic temperature cannot be defined when thesystem is submitted to a femtosecond laser because such a laser drives theelectronic population far from the Fermi-Dirac equilibrium distribution, where atemperature can be defined. The system returns to equilibrium conditions in a timelonger than the desorption time, of less than 1 ps. One of the possible ways thatpermits a withdrawal of the assumption of an electronic temperature is to use

528 D. Bejan 10

parameters dependent on the hot electrons nonequilibrium population as we did ina previous paper [11].

As a further extension, the desorption models should include supplementaryeffective PECs, in order to take into account the continuum of states in the metaland the interaction with the bath. A model using a third PEC associated with adamped state of the metal will be presented in the next paper.

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